Can noise, from a few seconds of low flying jet, arrive at my position all at once?

it was hard to ask this question in such a short title, sorry
This question has been in my mind for decades.
When I lived in Wales, the hilly terrain was used for training by RAF fighter jets which would cross a hill about 4k away, and do what
appeared to be a mock bombing run on our large white house at the end of the valley.
The noise was normally very loud, but roughly the same each time they did it, more or less.
But on occasions, as the jet approached in a very straight line towards me, the sound became so intense that it hurt the skin, it messed with my senses
even with my fingers in my ears, the air around my head appeared to be distorted, agitated, and turbulent, distorting the light,or so it looked.

I wondered if under certain conditions, jet height, approach speed, and flying directly at me, if it was possible, that all the sound made
over a period of say 4 seconds, arrived at my location simultaneously.
It is easy to draw such a scenario on graph paper for each second, but what about all the time in between
Is this scenario theoretically and mathematically possible?

Much easier to give a short title to this answer.
When the aircraft is travelling at the speed of sound (about 700 miles per hour) then as you suggest the sound it was making 4K away will arrive at the same time as the sound it is making now.
However this is not the only effect, as the aircraft pushes itself through the air at sonic or supersonic speeds it will create shock-waves.
These are regions where the air pressure changes very rapidly in a very short distance.
While the noise from a jet fighter at close range is truly appalling,
I suspect that it would be the sudden pressure pulse of the shock wave that would cause most of the effects you describe.

I agree with MBW, but would note that even if the jet's speed is sub-sonic (which it most likely is), if it flies directly at you energy of the power of the sound waves that reach you are much greater than if the jet is flying at an angle to you. For example at 80% the speed of sound the power of sound from a jet flying directly at you hear is 5 times the "normal " sound power. In general the power can be estimated from 1/(1-M), where M is the mach speed of the plane. Note that it this works only for M<1, and it tends to infinity at M=1: this is the cause of the sonic boom that you hear from a supersonic jet. I can imagine that under this condition your senses could be affected, causing the tingling and apparent distortion of sight.

Thanks to you both for replying.
The jets were most definitely sub sonic and low enough to see the pilots.

I am curious if the combined sound made over a period of a few seconds can arrive at my location all at the same time.

In other words, the sound made at 3 seconds flying time away from me, at 2, and at 1 second and much of the time segments in between.

The jets flew directly at me for the described effect to happen.
I have sketched this many times over the years, and it was one of the first things I tried to prove with my BBC home computer in 1983!.
I can sketch it again for you if I am still unclear.

Yes, that was precisely what I was trying to convey. If the plane is traveling at Mach 0.8 directly toward you, then every second you would receive 5 seconds worth of noise from the plane. That's what I meant about the sound of the jet having 5 times the power (i.e 5 times louder) than normal. You can see how this works by considering the following - I will use distance as measured in "sound-seconds," or "s-s," where one s-s = the distance traveled by sound in one second, which is about 1000 feet.

Let t = 0 be the time when the plane is 4 s-s away. Sound emitted at that time arrives at your ears 4 seconds later, or at t=4.

At t= 1 second the plane is 3.2 s-s away, and its sound from that point arrives at t= 1 + 3.2 = 4.2s.

At t= 2 the plane is 2.4 s-s away, and the sound emitted at that point arrives at t= 2+2.4 = 4.4s.

At t= 3 the plane is 1.6 s-s away, and the sound emitted at that point arrives at t= 3+1.6 = 4.6s.

At t= 4 the plane is 0.8 s-s away, and the sound emitted at that point arrives at t= 4+0.8 = 4.8s.

At t= 5 the plane is 0 s-s away, and the sound emitted at that point arrives at t= 5+0 = 5.0s.

Thus in the in the interval from t=4s to t= 5s you receive all the noise from the jet that was created from t=0s to t-5s. Clearly the intensity of the noise would be much greater than if the plane was flying in any direction other than towards you.

Yes, as already noted in my earlier post if speed = mach 1 (or greater)then sound energy just piles up on the nose of the moving plane. When it passes we perceive all that energy as a sonic boom. Yuo wouldn't hear rthe plane coming toward you - all you'd hear is the sonic boom as it goes past, followed by trailing noise as it speeds away.